Citations & acknowledgments

The happiest discovery

midi-sink set out to build Jaffer’s mathematical marbling into an instrument, then added two operators of its own: a wake behind the stylus and a Lamb–Oseen swirl driven by per-note pressure. Both turned out to be Aubrey Jaffer’s own published extensions of his closed-form family, arrived at here independently and found afterwards. The wake and swirl pages each carry their lineage note; the papers are below. Details were checked against the publications while writing this page, not recalled.

Mathematical marbling

1. Shufang Lu, Aubrey Jaffer, Xiaogang Jin, Hanli Zhao and Xiaoyang Mao, “Mathematical Marbling,” IEEE Computer Graphics and Applications, vol. 32, no. 6, pp. 26–35, 2012. doi:10.1109/MCG.2011.51 · author page. The closed-form drop, tine and exponential vortex; the inverse-lookup framework this whole engine is.
2. Aubrey Jaffer, “Oseen Flow in Paint Marbling,” arXiv:1702.02106, 2017. arxiv.org/abs/1702.02106 · stroke.pdf. An exact velocity field for Oseen (slow viscous) flow past a stylus and its use as a short-stroke marbling homeomorphism — the wake’s lineage.
3. Aubrey G. Jaffer, “The Lamb–Oseen Vortex and Paint Marbling,” arXiv:1810.04646, 2018. arxiv.org/abs/1810.04646 · vortex.pdf. The closed-form displacement pattern of a decaying Lamb–Oseen vortex — the swirl’s own paper (and the observation that real vortices are rare in marbling practice, because the Reynolds numbers needed want a far thinner size than marblers use).
4. Aubrey Jaffer, Mathematical Marbling — the marbling pages at MIT CSAIL: people.csail.mit.edu/jaffer/Marbling/. The mathematics, serpentine and bouquet patterns, dropping paint, transfer effects, and Pigment Transport in Paint Marbling (transfer.pdf), which analyses the Spanish wave and the Turkish moiré. The three animations there (Bouquet, Latte, Wave) are the ones the gallery’s tribute performance answers.

Classical fluid mechanics

5. Horace Lamb, Hydrodynamics, 6th edition, Cambridge University Press, 1932 (reprinted unaltered by Dover Publications, New York, 1945; Cambridge Mathematical Library, 1993). The two-dimensional doublet behind the wake and the vortex motion behind the swirl.
6. William John Macquorn Rankine, A Manual of Applied Mechanics, London: Charles Griffin, 1858. The combined vortex — rigid core, $1/r$ velocity outside — that the Rankine profile rotates by.

The electric medium’s operators

7. Ernst Florens Friedrich Chladni, Entdeckungen über die Theorie des Klanges, Leipzig, 1787. The figures: sand gathering on the nodal lines of a bowed plate — the picture the Chladni operator draws by stretching instead of gathering.
8. Geoffrey Ingram Taylor and Albert Edward Green, “Mechanism of the production of small eddies from large ones,” Proceedings of the Royal Society A, vol. 158, pp. 499–521, 1937. The cellular vortex whose stream function is the plate’s mode shape — the two-wave Chladni gesture, split into two exact shears.
9. Hassan Aref, “Stirring by chaotic advection,” Journal of Fluid Mechanics, vol. 143, pp. 1–21, 1984; Julio M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge University Press, 1989. Why iterating an exact flow winds the ink along its separatrices.
10. Boris V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Physics Reports, vol. 52, no. 5, pp. 263–379, 1979 (the standard map; its first statement in his 1969 Novosibirsk preprint). The Chirikov operator and Suzu’s kicked rotor.
11. John M. Greene, “A method for determining a stochastic transition,” Journal of Mathematical Physics, vol. 20, pp. 1183–1201, 1979. The residue method and the threshold $K_c = 0.971635$.
12. Jürgen Moser, “On invariant curves of area-preserving mappings of an annulus,” Nachrichten der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, pp. 1–20, 1962. The invariant curves that survive below the threshold — the KAM theorem’s map-side statement, the reason the ink slides along sheets before it shreds.
13. James D. Meiss, “Symplectic maps, variational principles, and transport,” Reviews of Modern Physics, vol. 64, no. 3, pp. 795–848, 1992. The island chains and the transport the canvas shows past the threshold.
14. S. I. Voropayev and Y. D. Afanasyev, Vortex Structures in a Stratified Fluid: Order from Chaos, Chapman & Hall, 1994; A. T. Chan and A. T. Chwang, “Unsteady singularities of Stokes’ flows in two dimensions,” International Journal of Engineering Science, 1995; Thierry Gallay and C. Eugene Wayne on the long-time asymptotics of the two-dimensional vorticity equation (2002, 2005). The viscous multipoles — the velocity fields the burst integrates in time by Jaffer’s method. The literature check behind the burst page’s wording was recorded before the page was written: the time-integrated displacement, elementary for $m \ge 2$, was not found stated in that form, and the page claims nothing beyond what it needed.

The synth’s acoustics

15. Arthur H. Benade, Fundamentals of Musical Acoustics, Oxford University Press, 1976 (Dover, 1990). The truncated cone with a mouthpiece holding the missing apex’s volume — the rule that put the saxophone’s peaks back on the integers.
16. Neville H. Fletcher and Thomas D. Rossing, The Physics of Musical Instruments, 2nd edition, Springer, 1998. The jet-edge flute, the reed and the lips as pressure-controlled valves, the flared bore’s inharmonic peaks.
17. Kevin Karplus and Alex Strong, “Digital synthesis of plucked-string and drum timbres,” Computer Music Journal, vol. 7, no. 2, pp. 43–55, 1983. The averaging loss and the round-trip law the hybrid string declares; the modal pluck in closed form is its spectrum.
18. Hal Chamberlin, Musical Applications of Microprocessors, Hayden, 1980 (2nd edition 1985). The state-variable filter that is Suzu’s one declared dissipation in the cell.
19. John W. Gordon and Julius O. Smith, “A sine generation algorithm for VLSI applications,” Proceedings of the International Computer Music Conference, pp. 165–168, 1985. The magic-circle oscillator — two shears per sample, $\det = 1$, exact in pitch — that is Suzu’s cell.
20. Loup Verlet, “Computer ‘experiments’ on classical fluids,” Physical Review, vol. 159, pp. 98–103, 1967. The integrator of the chain and the bore.

MIDI and the sampler’s format

21. MIDI Manufacturers Association and Association of Musical Electronics Industry, MIDI Polyphonic Expression (MPE) Specification, version 1.1 (document M1-100-UM), 14 April 2022; supersedes version 1.0 (RP-053, 2018). midi.org/mpe-midi-polyphonic-expression. Zones, the MPE Configuration Message (RPN 6), per-note channels, the default bend ranges — everything the implementation chart declares.
22. Decent Sampler by Dave Hilowitz (Decidedly Sound) — the .dspreset / .dslibrary format the sound engine’s sampler reads: decentsamples.com. What midi-sink reads of it, and what it bundles, is on the licensing page.

Software midi-sink is built with

• sokol_gfx and sokol-shdc — Andre Weissflog’s single-header graphics layer and shader cross-compiler: one shader source, Metal / HLSL / GLSL / GLES / WGSL.
• GLFW, Dear ImGui and libremidi in the desktop app; glm inside the core.
• Emscripten with Dawn’s emdawnwebgpu port for the web build; lil-gui for its settings panel.
• In the sound engine: miniaudio (David Reid), pugixml, miniz, and David Reid’s dr_wav and dr_flac; the Dan Tranh demo from the Versilian Community Sample Library (CC0).
• This site: Astro Starlight and KaTeX.

Thanks

To Professor Jaffer, first and always: for a family of equations generous enough to become an instrument, for publishing the two extensions we thought were ours, and for the method the burst only had to follow. To Ichisuke Fujioka, the Kanazawa suminagashi master whose floating ink this project watched before it wrote a line, and to the marblers of Turkey and Japan, whose craft this is a small, electric homage to. To the ROLI, Expressive E, Roland and Odisei engineers whose controllers shaped the input landscape; to Dave Hilowitz for a sampler format open enough to read; and to the Versilian Studios community for the zither that makes the first sound.